Figure 1.
A bifurcation diagram in terms of the $v$ component of the critical points as a function of the stratification parameter $s$ with fixed values $f =0.01$ and $r = 0.01$
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October 2018, 23(8): 3047-3070.
doi: 10.3934/dcdsb.2017196
On the scale dynamics of the tropical cyclone intensity
| 1. | Department of Earth and Atmospheric Sciences, Indiana University, Bloomington, IN 47405, USA |
| 2. | Department of Mathematics, Sichuan University, Sichuan Sheng, China |
This study examines the dynamics of tropical cyclone (TC) development in a TC scale framework. It is shown that this TC-scale dynamics contains the maximum potential intensity (MPI) limit as an asymptotically stable point for which the Coriolis force and the tropospheric stratification are two key parameters responsible for the bifurcation of TC development. In particular, it is found that the Coriolis force breaks the symmetry of the TC development and results in a larger basin of attraction toward the cyclonic (anticyclonic) stable point in the Northern (Southern) Hemisphere. Despite the sensitive dependence of intensity bifurcation on these two parameters, the structurally stable property of the MPI critical point is maintained for a wide range of parameters.
Mathematics Subject Classification:
Primary: 76U05, 76E09; Secondary: 58F15, 58F17.
Citation:
Chanh Kieu, Quan Wang. On the scale dynamics of the tropical cyclone intensity. Discrete & Continuous Dynamical Systems - B,
2018, 23
(8)
: 3047-3070.
doi: 10.3934/dcdsb.2017196
![]()
References:
| [1] |
B. R. Brown and G. J. Hakim,
Variability and predictability of a three-dimensional hurricane in statistical equilibrium, J. Atmos. Sci., 70 (2013), 1806-1820.
doi: 10.1175/JAS-D-12-0112.1. |
| [2] |
G. H. Bryan and R. Rotunno,
The maximum intensity of tropical cyclones in axisymmetric numerical model simulations, Mon. Wea. Rev, 137 (2009), 1770-1789.
doi: 10.1175/2008MWR2709.1. |
| [3] |
J. G. Charney and A. Eliassen,
On the growth of the hurricane depression, J. Atmos. Sci, 21 (1964), 68-75.
doi: 10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2. |
| [4] |
K. A. Emanuel,
A statistical analysis of tropical cyclone intensity, Monthly Weather Review, 128 (2000), 1139-1152.
doi: 10.1175/1520-0493(2000)128<1139:ASAOTC>2.0.CO;2. |
| [5] |
K. A. Emanuel,
An air-sea interaction theory for tropical cyclones. part i: Steady-state maintenance, J. Atmos. Sci, 43 (1986), 585-605.
doi: 10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2. |
| [6] |
K. A. Emanuel,
The Maximum Intensity of Hurricanes, J. Atmos. Sci, 45 (1986), 1143-1155.
doi: 10.1175/1520-0469(1988)045<1143:TMIOH>2.0.CO;2. |
| [7] |
M. Ferrara,, F. Groff, Z. Moon, K. Keshavamurthy, S. M. Robeson and C. Kieu, Large-scale control of the lower stratosphere on variability of tropical cyclone intensity, Geophys. Res. Lett., 44 (2017), 4313-4323. |
| [8] |
G. J. Hakim,
The mean state of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 68 (2011), 1364-1376.
doi: 10.1175/2010JAS3644.1. |
| [9] |
G. J. Hakim,
The variability and predictability of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 70 (2013), 993-1005.
doi: 10.1175/JAS-D-12-0188.1. |
| [10] |
K. A. Hill and G. M. Lackmann,
The impact of future climate change on TC intensity and structure: A downscaling approach, Journal of Climate, 24 (2011), 4644-4661.
doi: 10.1175/2011JCLI3761.1. |
| [11] |
C. Kieu,
Hurricane maximum potential intensity equilibrium, Q.J.R. Meteorol. Soc., 141 (2015), 2471-2480.
doi: 10.1002/qj.2556. |
| [12] |
C. Kieu and Z. Moon,
Hurricane intensity predictability, Bull. Amer. Meteo. Soc., 97 (2016), 1847-1857.
doi: 10.1175/BAMS-D-15-00168.1. |
| [13] |
C. Kieu, H. Chen and D. L. Zhang, An examination of the pressure-wind relationship for intense tropical cyclones, Wea. and Forecasting, 25 (2010), 895-907. |
| [14] |
Y. Liu, D.-L. Zhang and M. K. Yau, A Multiscale Numerical Study of Hurricane Andrew Part Ⅱ: Kinematics and Inner-Core Structures, J. Atmos. Sci, 127 (1999), 2597-2616. |
| [15] |
T. Ma, Topology of Manifolds Science Press, Beijing. , 2007. |
| [16] |
J. W. Milnor,
Topology from the Differentiable Viewpoint Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va. 1965. |
| [17] |
Y. Ogura and N. A. Phillips, Scale analysis of deep and shallow convection in the atmophere, J. Atmos. Sci, 19 (1962), 173-179. |
| [18] |
R. Rotunno and K. A. Emanuel, An airsea interaction theory for tropical cyclones. part ii: Evolutionary study using a nonhydrostatic axisymmetric numerical model, J. Atmos. Sci, 44 (1987), 542-561. |
| [19] |
W. Shen, R. E. Tuleya and I. Ginis, A sensitivity study of the thermodynamic environment on GFDL model hurricane intensity: Implications for global warming, J. Climate, 13 (2000), 109-121. |
| [20] |
R. K. Smith, G. Kilroy and M. T. Montgomery,
Why do model tropical cyclones intensify more rapidly at low latitudes, Journal of the Atmospheric Sciences, 72 (2015), 1783-1840.
doi: 10.1175/JAS-D-14-0044.1. |
| [21] |
R. Wilhelmson and Y. Ogura,
The pressure perturbation and the numerical modeling of a cloud, Journal of the Atmospheric Sciences, 29 (1972), 1295-1307.
doi: 10.1175/1520-0469(1972)029<1295:TPPATN>2.0.CO;2. |
| [22] |
H. E. Willoughby,
Forced secondary circulations in hurricanes, J. Geophys. Res, 84 (1979), 3173-3183.
doi: 10.1029/JC084iC06p03173. |
| [23] |
H. E. Willoughby,
Gradient Balance in Tropical Cyclones, J. Atmos. Sci., 47 (1990), 265-274.
doi: 10.1175/1520-0469(1990)047<0265:GBITC>2.0.CO;2. |
show all references
References:
| [1] |
B. R. Brown and G. J. Hakim,
Variability and predictability of a three-dimensional hurricane in statistical equilibrium, J. Atmos. Sci., 70 (2013), 1806-1820.
doi: 10.1175/JAS-D-12-0112.1. |
| [2] |
G. H. Bryan and R. Rotunno,
The maximum intensity of tropical cyclones in axisymmetric numerical model simulations, Mon. Wea. Rev, 137 (2009), 1770-1789.
doi: 10.1175/2008MWR2709.1. |
| [3] |
J. G. Charney and A. Eliassen,
On the growth of the hurricane depression, J. Atmos. Sci, 21 (1964), 68-75.
doi: 10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2. |
| [4] |
K. A. Emanuel,
A statistical analysis of tropical cyclone intensity, Monthly Weather Review, 128 (2000), 1139-1152.
doi: 10.1175/1520-0493(2000)128<1139:ASAOTC>2.0.CO;2. |
| [5] |
K. A. Emanuel,
An air-sea interaction theory for tropical cyclones. part i: Steady-state maintenance, J. Atmos. Sci, 43 (1986), 585-605.
doi: 10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2. |
| [6] |
K. A. Emanuel,
The Maximum Intensity of Hurricanes, J. Atmos. Sci, 45 (1986), 1143-1155.
doi: 10.1175/1520-0469(1988)045<1143:TMIOH>2.0.CO;2. |
| [7] |
M. Ferrara,, F. Groff, Z. Moon, K. Keshavamurthy, S. M. Robeson and C. Kieu, Large-scale control of the lower stratosphere on variability of tropical cyclone intensity, Geophys. Res. Lett., 44 (2017), 4313-4323. |
| [8] |
G. J. Hakim,
The mean state of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 68 (2011), 1364-1376.
doi: 10.1175/2010JAS3644.1. |
| [9] |
G. J. Hakim,
The variability and predictability of axisymmetric hurricanes in statistical equilibrium, J. Atmos. Sci, 70 (2013), 993-1005.
doi: 10.1175/JAS-D-12-0188.1. |
| [10] |
K. A. Hill and G. M. Lackmann,
The impact of future climate change on TC intensity and structure: A downscaling approach, Journal of Climate, 24 (2011), 4644-4661.
doi: 10.1175/2011JCLI3761.1. |
| [11] |
C. Kieu,
Hurricane maximum potential intensity equilibrium, Q.J.R. Meteorol. Soc., 141 (2015), 2471-2480.
doi: 10.1002/qj.2556. |
| [12] |
C. Kieu and Z. Moon,
Hurricane intensity predictability, Bull. Amer. Meteo. Soc., 97 (2016), 1847-1857.
doi: 10.1175/BAMS-D-15-00168.1. |
| [13] |
C. Kieu, H. Chen and D. L. Zhang, An examination of the pressure-wind relationship for intense tropical cyclones, Wea. and Forecasting, 25 (2010), 895-907. |
| [14] |
Y. Liu, D.-L. Zhang and M. K. Yau, A Multiscale Numerical Study of Hurricane Andrew Part Ⅱ: Kinematics and Inner-Core Structures, J. Atmos. Sci, 127 (1999), 2597-2616. |
| [15] |
T. Ma, Topology of Manifolds Science Press, Beijing. , 2007. |
| [16] |
J. W. Milnor,
Topology from the Differentiable Viewpoint Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va. 1965. |
| [17] |
Y. Ogura and N. A. Phillips, Scale analysis of deep and shallow convection in the atmophere, J. Atmos. Sci, 19 (1962), 173-179. |
| [18] |
R. Rotunno and K. A. Emanuel, An airsea interaction theory for tropical cyclones. part ii: Evolutionary study using a nonhydrostatic axisymmetric numerical model, J. Atmos. Sci, 44 (1987), 542-561. |
| [19] |
W. Shen, R. E. Tuleya and I. Ginis, A sensitivity study of the thermodynamic environment on GFDL model hurricane intensity: Implications for global warming, J. Climate, 13 (2000), 109-121. |
| [20] |
R. K. Smith, G. Kilroy and M. T. Montgomery,
Why do model tropical cyclones intensify more rapidly at low latitudes, Journal of the Atmospheric Sciences, 72 (2015), 1783-1840.
doi: 10.1175/JAS-D-14-0044.1. |
| [21] |
R. Wilhelmson and Y. Ogura,
The pressure perturbation and the numerical modeling of a cloud, Journal of the Atmospheric Sciences, 29 (1972), 1295-1307.
doi: 10.1175/1520-0469(1972)029<1295:TPPATN>2.0.CO;2. |
| [22] |
H. E. Willoughby,
Forced secondary circulations in hurricanes, J. Geophys. Res, 84 (1979), 3173-3183.
doi: 10.1029/JC084iC06p03173. |
| [23] |
H. E. Willoughby,
Gradient Balance in Tropical Cyclones, J. Atmos. Sci., 47 (1990), 265-274.
doi: 10.1175/1520-0469(1990)047<0265:GBITC>2.0.CO;2. |
Figure 2.
A bifurcation diagram in terms of the $v$ component of the critical points as a function of the Coriolis parameter $f$ with fixed values $s = 0.1, r = 0.01$
Figure 3.
a) Flow trajectories for four different initial points in the phase space of $(u, v, b)$ that represent an incipient weak anticyclonic vortex (-0.1, -0.1, 0.1) (red); a mature TC near the MPI equilibrium with a weak warm core (-1, 1, 0.5) (cyan); a mature TC with intensity significant above the MPI equilibrium limit (-1, 1.4, 1) (green); and a mature TC near the MPI equilibrium limit with too weak low level convergence (-0.1, 1, 1) (blue) for the case of $f = 0.05$ ; (b) Time series of $v$ during the entire simulation; and (c)-(d) Similar to (a)-(b) but for the case of $f=0$
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